Oscillators are vital elements of clocks and have many important applications in telecommunications, remote sensing (e.g. radar), and signal processing. An oscillator is a device that supplies a measurable output. This output can take, for example, the form of the deflection of a mechanical dial or the voltage across a pair of electrical terminals. An oscillator regulates the value of its output as a function of time such that it is periodic (with respect to time). The inverse of the output's temporal period is known as the oscillator's frequency.
Performance, Requirements, and Fundamental Anatomy of an Oscillator
Performance:
An oscillator's fitness of purpose, with respect to any particular application, is determined by the extent to which its frequency remains constant as time progresses. Oscillators that exhibit the most stable output frequencies are the ones, and the only ones, that can enable the most exacting applications, and they are thus valuable.
A mathematical phase can be associated with an oscillator's output, where an increase of 2π in the value of this phase is associated with each new period of the oscillator's output. The phase can be notional compared to that of a corresponding ideal oscillator, whose phase accumulates perfectly linearly with respect to time. The difference in the phases of these two oscillators as a function of time, particularly when transformed into the so-called Fourier domain, provides an alternative way of viewing and characterising the real oscillator's frequency stability.
Several established measures of performance, well known to experts in the art of oscillator characterization, are used to quantify an oscillator's frequency stability. These measures have been reviewed by Rutman and Walls [‘Characterization of Frequency Stability In Precision Frequency Sources’, J. Rutman and F. L. Walls, Proceedings of the IEEE, vol. 79, pp. 952-960 (1991)] and by Stein [‘Frequency and Time—Their Measurement and Characterization’, S. R. Stein, pp. 191-416, in ‘Precision Frequency Control’, edited by E. A. Gerber and A. Ballato, Academic Press, New York (1985)], as well as in many other technical articles and monographs.
Two Measures of performance that have gained particularly wide industrial acceptance are:    (i) The oscillator's ‘fractional frequency stability’ defined mathematically as its square-root Allan variance, also known as its Allan deviation; this measure is a function of the temporal sampling interval.and    (ii) The oscillator's ‘phase-noise spectral density’, defined mathematically as the one-sided spectral density of the oscillator's phase fluctuations (with respect its ideal equivalent); this measure is a function of the offset frequency with respect to oscillator's average, or so-called ‘carrier’ frequency.
Note that each of these two measures of performance is a curve, i.e. a function whose value varies with the value of the function's argument, as opposed to being a single numerical value. With either of these measures, for a given value of its argument (i.e. for a given value of the temporal sampling interval in the case of the fractional frequency stability, or for a given value of the offset frequency, in case the phase-noise spectral density), the lower the measure the greater the oscillator's frequency stability. An oscillator of high performance with regard to it frequency stability is one that exhibits a low fractional frequency stability over a range of temporal sampling intervals or one that exhibits a low phase-noise spectral density, over a range of frequency offsets.
Requirements:
Beyond its frequency-stability performance, an oscillator has various other properties that can affect its utility in particular applications. For a start, an oscillator has a certain mass and occupies a certain amount of space. An oscillator requires the supply of various resources to it and also the keeping of it in an environment whose qualities should allow the oscillator to attain the level of frequency stability that is required of it. All oscillators require both an adequate supply of power and an adequate means of discharging waste heat. Certain oscillators can only operate over limited ranges of temperature. Many other environmental parameters can affect the oscillator's operation, and fluctuations in them may affect the oscillator's frequency stability. These parameters include: the magnitude and direction of either ambient or deliberately applied electric or magnetic fields; the humidity; the intensity of ionising radiation to which the oscillator is exposed; mechanical acceleration (caused by the movement/vibration of the platform to which the oscillator is attached) and acoustic noise.
Fundamental Anatomy:
Every oscillator contains, in conjunction with peripheral supporting equipment, the following three core functional elements:    (A) a means of defining a particular (absolute) frequency; this ‘frequency reference’ is typically associated with a ‘resonance’, as exhibited by some or other form of ‘resonator’;    (B) a means for ensuring that the oscillator oscillates at the reference frequency; this process of regulation or ‘locking’ is typically achieved through some of other form of ‘interference’ or ‘feedback’.    (C) a means of sustaining the oscillator's oscillation (to prevent it from decaying through unavoidable dissipative processes); this energy-supplying function is typically accomplished by some or other form of ‘amplifier’.
The frequency reference associated with any individual oscillator need not be perfectly reproducible but it should, by definition, be sufficiently constant with respect to time. This in turn requires that those physical properties of the resonator whose resonance defines the frequency reference should remain sufficiently constant with time. The present invention proposes a new type of oscillator with operating frequencies in or around the broadly interpreted ‘microwave’ region, i.e. between a few hundred MHz and several hundred GHz. Here, to provide some technological context for the invention, and to appreciate its significance, some relevant existing microwave oscillators and associated components are briefly reviewed.
Passive Electromagnetic Resonators and Whispering-Gallery Modes
Many successful designs of microwave oscillator incorporate a passive electromagnetic (e.m.) resonator to supply the frequency-reference function [(A) above], where the resonator supports one or several resonant, electromagnetic modes exhibiting high quality factors. Here, the resonator functions as an essentially passive, linear (either 1- or 2-port) device, as can be modelled with scattering parameters (e.g. S21). Because of their regular mention within the descriptions that follow, the quality factors of e.m. modes shall often be denoted simply by the (italicised) letter ‘Q’. The frequency and Q of an e.m. mode is determined by the dimensions, shape and electromagnetic properties (viz. electric permittivity, magnetic susceptibility, surface conductance . . . ) of the resonator's constituent materials. The oscillator's frequency is regulated, by one of several available means, to lie at or close to the centre frequency of a particular high-Q mode. One says that the oscillator ‘runs on’, or is ‘locked to’ the mode. Besides the resonator itself, the oscillator requires a means of amplification to sustain its oscillation and also a means for locking its operating frequency to the mode on which it is intended to run. The most stable microwave oscillators based on high-Q resonators were recently reviewed by A. G. Mann. [‘Ultrastable Cryogenic Microwave Oscillators’, A. G Mann, pp. 37-66, in ‘Frequency Measurement and Control, Advanced Techniques and Future Trends’, Edited by A. N. Luiten, Topics in Applied Physics Vol. 79, Springer-Verlag 2000).] Many (though not all) of them are locked to electromagnetic mode of so-called ‘whispering-gallery’ type.
Without any further associated qualifications, the word ‘dielectric’, is used below to denote materials that have a complex electric permittivity, at the frequency of the electromagnetic wave to which the material is exposed, whose real part is greater than that of free space and whose complex part is several orders of magnitude smaller than its real part; in other words, a dielectric is, by default, a ‘good’ one, in the sense of it having a low loss tangent. Mica, silica, rutile, polystyrene and p.t.f.e. are examples of such dielectrics at microwave frequencies. A sinusoidal electromagnetic wave is characterized by its associated wavelength, λ, which is in general a function of the wave's frequency, polarization and direction of propagation. Through unbounded regions of a (good) dielectric, such a wave can propagate over distances equal to many wavelengths before suffering significant attenuation.
The external surface of a dielectric body that is surrounded by free space defines an extended two-dimensional electromagnetic interface. If the body's form is suitably convex, this interface can support a distinct class of electromagnetic waves known as ‘whispering gallery’ (henceforth ‘WG’) waves or modes, as were first analysed by Lord Rayleigh in the case of analogous acoustical systems [‘The problem of the whispering gallery’, J. W. S. Rayleigh, Philosophical Magazine, Vol. 20, pp. 1001-4 (1910)]. The WG waves flow tangential to the interface and are predominantly confined to the dielectric side of it, in a layer of thickness √{square root over (λR)} that lies immediately below the surface; here, λ is the wavelength of the equivalent freely-propagating wave in an infinite region of the same dielectric and R is the surface's appropriate radius of curvature. The mechanism of confinement can be considered as the radio- and/or microwave-frequency equivalent of what in ray optics is known as ‘total internal reflection’. Within the ray-optics description, the inverse sine of the WG wave's angle of incidence at the interface is always greater than the dielectric's refractive index (equal to the square root of the ratio of the dielectric's relative electric permittivity to that of free space). Since no corresponding transmitted (refracted) wave propagating in free space can satisfy Snell's law, the WG wave is totally reflected back into the dielectric. Outside of the dielectric structure, the wave is evanescent as opposed to propagating, where its amplitude (and also its field-energy density) decays exponentially with distance away from the interface. All these features are most pronounced when the wavelength λ is small compared to the convex dielectric body's characteristic radius of curvature R. If the dielectric body is compact or at least ‘closed’, in the sense that it confines WG waves to run in closed loops, then multiple interference will select a discrete spectrum of frequencies to each of which a particular whispering-gallery standing wave or ‘mode’ is associated.
Finite dielectric bodies whose geometric shapes have rotational symmetry can be made relatively easily with commonly available tools and established methods of fabrication. Most work on electromagnetic WG-modes to date has concerned dielectric bodies with cylindrical symmetry in the form of spheres, or cylinders, or more general ‘solids of rotation’ such as rings or toroids. Such dielectric bodies will support whispering-gallery modes that exhibit (discrete) rotational symmetry. The rotational symmetry significantly aids, moreover, in the mathematical analysis and electromagnetic modelling of these WG modes as they can then be quite accurately represented in terms of special functions (such as Bessel functions and their ‘modified’ variants) or at least relatively compact series-expansions thereof. However, as pointed out below, rotational symmetry, though often convenient, is not strictly necessary for supporting or analysing WG-modes.
If the electric permittivity tensor of the dielectric material out of which such a body is made is anisotropic, as is the case with many crystalline dielectrics whose losses are desirably low, yet the tensor exhibits a rotational symmetry about a particular axis, as is also often the case with the same, this axis can be oriented parallel to the body's geometric axis of rotational symmetry such that the electric permittivity of the space within and about the dielectric body exhibits continuous rotational symmetry. The requisite alignment of the crystal can be accomplished by either viewing it optically through crossed polarizers or using x-ray diffraction. Monocrystalline sapphire is an example of such a material; its c-axis is oriented parallel to the cylindrical axis of the dielectric body that is comprises it. The dielectric loss of sapphire at microwave frequencies, especially at liquid-helium temperatures, is, moreover, extremely low.
Dielectric resonators incorporating rings, cylinders or pucks of high-purity monocrystalline sapphire have been used in electromagnetic resonators at cryogenic temperatures to support WG modes at microwave frequencies exhibiting unloaded Qs in excess of 109. Microwave oscillators built around such resonators have excellent frequency stability. [See ‘Improved cryogenic sapphire oscillator with exceptionally high frequency stability’, S. Chang et al, Electronics. Letters 36, 480-481 (2000); ‘Cryo-cooled sapphire oscillator with ultra-high stability’, G. J. Dick et al, Proceedings of 1998 IEEE International Frequency Control Symposium, pp. 528-533 (1998); ‘A cryogenic open-cavity sapphire reference oscillator with low spurious mode density’, P.-Y. Bourgeois, IEEE Transanctions on Ultrasonics. Ferroelectrics and Frequency Control, vol. 51, pp. 1232-1239 (2004).] Here, the sapphire resonator is maintained at a temperature lying conveniently above 4.2 Kelvin, where a frequency-versus-temperature turnover point is located. The turnover point can be precisely sat upon through a temperature-control servo loop that employs a resistive heater, with no need to pump on the helium of bath. It should be noted here, and as Bourgeois et al explicitly have demonstrated, a reflective electromagnetic shield or cavity placed around the dielectric ring is not necessary for attaining attractively high Qs.
The whispering-gallery modes can be classified through their associated electromagnetic field configurations. Assuming rotational symmetry, the modes can be broadly divided into two classes: quasi-transverse-magnetic (WGH) modes, whose magnetic field lines lie approximately orthogonal to the ring's cylindrical axis, and quasi-transverse-electric (WGE) modes whose electric field lines lie approximately normal to the mode's cylindrical axis. The azimuthal mode order, ‘n’, of a WG mode equals the number of full waves made by the mode's field pattern around the perimeter of its supporting ring. The WG mode's axial and radial mode orders, ‘a’ and ‘r’, respectively, equal the number of nodes in the mode's field pattern along these two respective directions. A ‘fundamental’ WG mode has no nodes in either of these two directions. In general, a WG mode can be identified using the notation WGHn,r,a for quasi-transverse-magnetic (WGH) modes and WGEn,r,a for quasi-transverse-electric modes. Without external perturbations (from coupling probes, for example) each WG is doubly degenerate; this degeneracy can be associated with two travelling waves circulating in opposite directions around the ring; alternatively, the doublet can be regarded as comprising two otherwise identical standing-wave WG modes whose azimuthal phase differs by 90°.
In general, the existence and form of a given whispering-gallery mode is robust to isolated defects, such as the odd inclusion or void within the dielectric body, or the odd surface scratch or chipped edge on the body's surface, provided the dimensions of these defects are small compared with the mode's associated wavelength λ (within the dielectric). Though the form of the whispering-gallery mode may remain qualitatively the same, its quality factor Q can be significantly degraded by even small defects through their scattering or absorption of energy from the WG mode.
The vast majority of the electromagnetic WG modes that have been studied to date have been planar in form. Planarity is not strictly essential to WG modes and indeed, non-planar whispering-gallery modes can be supported by finite cylinders and ‘bottle-shaped’ solids of rotation; as have recently been analysed by Sumetsky. [‘Whispering-gallery-bottle microcavities: the three-dimensional etalon’, M. Sumetsky, Optics Letters, Vol. 29, pp. 8-10 (2004)].
Furthermore, rotational symmetry of the dielectric body itself is not necessary for supporting ‘generalized’ whispering-gallery electromagnetic modes, which exhibit the same desirable features concerning confinement, low evanescent leakage and (thus) high Q, as their rotationally symmetric relatives. These generalized WG mode can, moreover, offer significant advantages and design flexibility with regard to controlling how they couple electromagnetically to surrounding structures (such as coupling probes and ‘launchers’) located at a particular azimuthal positions. S. Ancey et al have, for example, analysed whispering-gallery modes in dielectric bodies of elliptical shape [‘Whispering-gallery modes and resonances of an elliptic cavity’, S Ancey, A Folacci and P Gabrielli, Journal of Physics A (Mathematical and General), Vol. 34, pp 1341-1359 (2001)].
To support a generalized WG mode, the external surface of the dielectric body should in general contain a closed, convex ‘band’, where the band's curvature in its ‘long’ or generalized-azimuthal direction should be small enough at all positions around the band to ensure total internal reflection and (thus) sufficiently suppress evanescent leakage and radiation losses. Furthermore, the variation in the azimuthal curvature around the band needs to be sufficiently limited to avoid anomalously large leakage and losses due to a phenomenon known are ‘chaos assisted tunnelling’, as for example has been discussed by Nöckel and Stone [‘Ray and wave chaos in asymetric resonator optical cavities’, J. U. Nöckel, and A. D. Stone, Nature. 385, pp. 45-47 (1997)]. Provided this and related wave-chaotic phenomena (such as so-called ‘dynamical localization’) are taken in to account, non-rotationally symmetric dielectric bodies that support generalized whispering-gallery modes with both high Qs and advantageous coupling features can be rationally designed and constructed.
Oscillator Loops and Locking Configurations
In connection with (B) stated above, the various means through which an oscillator can be compelled to run at the frequency that is defined by its frequency reference (as embodied by a resonator) are reviewed here.
The simplest resonator-based oscillators comprise a passive resonator [i.e. (A) above] and an amplifier [i.e. (C) above] that are connected directly together by cables in a loop to form what is known as a free-running loop oscillator. The phase-noise performance of such oscillators has been by considered quantitatively by Leeson [‘A Simple Model of Feedback Oscillator Noise Spectrum’. D. B. Leeson, Proceedings of the IEEE, vol. 54, pp. 329-330 (1966)] and more recently by Everard in somewhat more detail [‘Fundamentals of RF Circuit Design with Low Noise Oscillators’, by J. Everard, John Wiley & Sons Ltd. (2001)]. Through such considerations, it is known how a loop oscillator's phase noise depends on its operating parameters. With regard to attaining low phase noise:    (i) the mode supported by the oscillator's resonator should have an unloaded Q that is as high as possible;    (ii) the mode's centre frequency should be as constant as possible;    (iii) the oscillator's feedback power, which flows through the resonator, should be as large as possible;    (iv) the thermal noise generated by the amplifier, as characterized by its finite noise temperature, as well as that generated by other parts of the oscillator's loop, should be as low as possible; thermal noise can be reduced by reducing the temperature;    (v) the flicker noise in the oscillator's sustaining amplifier should be as low as possible;    (vi) the electromagnetic properties of the loop's interconnecting cables, i.e. their phase length and loss, should be as stable as possible—lest fluctuations in the cables come to pull the loop oscillator's frequency.
The optimisation of a practical free-running loop oscillator involves judicious trade-offs between the above parameters. More complicated designs of microwave oscillator, incorporating additional components within various servo loops, attempt to suppress or circumvent the free-running loop oscillator's sensitivities to fluctuations in certain of them.
So-called Pound-stabilized loop (PSL) oscillators [see ‘A High Stability Microwave Oscillator Based on Sapphire Loaded Superconducting Cavity’, by A. J. Giles et al, Proceedings of the 43rd Annual Symposium on Frequency Control, pp. 89-93 (1989)] have been built around cryogenic sapphire whispering-gallery-mode resonators maintained at cryogenic (often liquid-helium) temperatures within a refrigerator (i.e. a so-called ‘cryostat’). Within the bandwidth of such an oscillator's Pound stabilizer, the frequency instabilities that would otherwise be introduced by either the loop amplifier's phase noise and/or by fluctuations in the loop's interconnecting cables, or both, are compensated.
In contrast to the present invention proposed below, it can be remarked that a PSL oscillator is a spatially extended system: at least two microwave lines, each typically greater than one meter in length, are required to connect the cryogenic resonator to the room-temperature section of the oscillator's loop. In reality, each line comprises a series of semi-rigid microwave cables that are connect by feedthroughs between the cryostat's different chambers and sections. To achieve frequency stabilities at the 1×10−14 level, several auxiliary cables and sensors, supporting the control of the resonator's temperature, the so-called Pound servo, and well as loop-power regulation [see, for example, ‘Latest results of the U.W.A. cryogenic sapphire oscillator’, A. N. Luiten et al, Proceedings of IEEE International 49th Frequency Control Symposium, pp. 433-437 (1995)] are all required to be wired into the cryostat. Pound stabilization does suppress, within a finite bandwidth, the phase (thus frequency) shifts associated with mechanical vibrations and/or temperature fluctuations along the microwave lines. Residual amplitude modulation, causing offsets in the Pound servo's d.c.˜error signal remains a problem, however.
Despite their configurational complexity, PSL oscillators have demonstrated their worth in several demanding applications. They have been used as ‘flywheels’ for cold-atom frequency standards [see ‘A high stability atomic fountain clock using a cryogenic sapphire interrogation oscillator’, A. G. Mann et al, Proceedings of 1998 IEEE International Frequency Control Symposium, Pasadena, Calif., USA, pp. 13-17 (1998)], as reference oscillators for (close-in) phase-noise measurements [see ‘Microwave Frequency Discriminator with a Cryogenic Sapphire Resonator for Ultra-Low Phase Noise’, G. J. Dick & D. G. Santiago, Proc. of 6th European Frequency and Time Forum, Noodwijk, The Nederlands, 17-19 March 1992, pp. 35-39 (1992)], or in fundamental-physics experiments testing, for example, Lorentz invariance [see ‘Tests of Lorentz Invariance using a Microwave Resonator’, P. Wolf et al, Physical Review Letters, vol. 90, pp. 060402 (2003)].
Solid-State Masers
In the designs of oscillators considered above, the means through which the oscillator's oscillation is sustained [(B) above] takes the form of a conventional microwave amplifier, typically based on semiconductor technology, and typically operating at room temperature. A wholly different means of amplification is Microwave Amplification by Stimulated Emission of Radiation, which is based on atomic (also often described as ‘quantum-electronic’) principles. The term ‘maser’ (as both a noun and adjective) shall henceforth be generally used to refer to it; the gerundive ‘mas(er)ing’ shall also on occasions be used to refer to those physical entities that participate in the ‘maser action’—to distinguish them from other's that don't.
The maser phenomemon can be realized in both solid-state systems, where the masering atoms (or ions) reside in condensed matter, and also in more rarified systems, where the masering atoms, in the form of propagating beams or clouds, reside in what is otherwise a vacuum. With regard to the former, solid-state masers have been reviewed pedagogically by Siegman [‘Microwave Solid-state Masers’, A. E. Siegman, McGraw-Hill (1964)]. Here, the masing medium is a dielectric solid, most often a crystal, containing a distribution (a so-called ‘solid dilution’) of paramagnetic ions. The frequencies that are associated with transitions between the electronic-spin states, or ‘levels’, of these paramagnetic ions typically lie in the microwave region (i.e. GHz). With a few exceptions, maser action in solids requires operation at liquid-helium temperatures. The most comprehensively studied and applied solid-state maser system to date has been (artificial) ‘pink ruby’, that is, crystalline sapphire doped with chromium Cr3+ ions, at a substitutive concentration on the order of 1 part per thousand. This system has been used to realize low-noise, sufficiently wide-band amplifiers for boosting the power of weak microwave signals in telecommunication and astronomy. [See, for example, ‘Solid State Masers’, N. Bloembergen, pp. 396-429 (Chapter IX) in ‘Progress in Low Temperature Physics’, edited by C. J. Gorter, vol. 111, North-Holland (1961).]
The strength of a paramagnetic transition between two levels in so-called ‘free-spin units’ is equal to the ratio, expressed as a fraction, between the said strength and that of the equivalent ‘strongly allowed’ transition between the upper (m=−½) and lower (m=+½) spin levels of a free electron (S=½), where these two free-spin levels have been separated by an applied static magnetic field. Maser action requires that both its so-called ‘pump’ and ‘signal’ transitions are sufficiently strong, i.e. a sufficiently large fraction of a free-spin unit. Except in a few specific systems, which shall be subsequently mentioned, this generally requires that the masering paramagnetic ions be exposed to a d.c. magnetic ‘bias’ field that splits and mixes, quantum mechanically, the ‘zero-field’ quantum states that the ions would otherwise have. In addition, the application of a magnetic field of a judiciously chosen strength and orientation (relative to the crystal's axes), allows the maser's signal and pump transitions to be ‘engineered’ with regard to their strengths, polarizations, and frequencies [see Siegman, already referenced above.]
Dick et al [‘Development of the Superconducting Cavity Maser’, G. J. Dick and D. M. Strayer, Proceedings of 38th Annual Frequency Control Symposium, pp. 435-446 (1984); ‘Ultra-Stable Performance of the Superconducting Cavity Maser’, G. J. Dick, and R. T. Wang, IEEE Transactions on Instrumentation and Measurement, vol. 40, pp.174-177 (1991)] developed a ‘Superconducting Cavity Maser Oscillator’ (SCMO), operating at a temperature near 1.6 K, that exhibited extremely good frequency stability (Allan deviation of 4–5×10−15 for sampling intervals between 1 and 1000 s) and exceptionally low (flicker-) phase noise (−80 dBc/f3, where the frequency f is in Hz). Dick et al's oscillator incorporated a solid-state ruby maser amplifier. This amplifier was maintained at a cryogenic temperature and its operation required an applied d.c. magnetic ‘bias’ field. With regard to the present proposed invention, it should be, noted that the maser amplifier within the SCMO was physically separated from the oscillator's passive frequency-defining resonator cavity. This physical separation was in fact an essential feature of the SCMO: the resonator's high Q, and hence the oscillator's frequency stability, would have been severely degraded, due to the trapping of magnetic flux in the walls of the oscillator's superconducting cavity (made of lead), had the cavity been exposed to the same magnetic field necessary to bias the ruby crystal. In other words, the amplifier and resonator of Dick et al's SCMO could not operate in the same region of space.
In contrast to a typical embodiment of the Pound-stabilized loop oscillator, however, the SCMO's all-cryogenic loop was relatively more compact (centimeters as opposed to meters) and benefited from the uniformity and stability of temperature provided by its wholly cryogenic environment. By dint of the ruby maser's combined low thermal and low flicker noise, the phase noise exhibited by the SCMO was superior to that which could have been achieved by swapping the maser for a conventional microwave (e.g. ‘GaAsFET’ or ‘HEMT’) amplifier, even if operated at a cryogenic temperature. The SCMO offered several other advantages that are shared with the present invention proposed below, which shall be stated in due course.
For completeness, and as shall be relevant to the description of the present invention's first experimental embodiment, it is noted here that there exist a few solid-state systems that enable maser action without the application of a d.c. magnetic bias field. Sapphire that is substitutively doped with Fe3+ ions, or sumarium sulphate doped with Gd3+ ions, provide two examples (there are several others). Such so-called ‘zero-field masers’. have been reviewed by Bogle and Symmons [‘Zero-Field Masers’, G. S. Bogle and H. F. Symmons, Australian Journal of Physics, 12, pp. 1-20 (1959)].
Atomic Maser Oscillators
If the linewidth of a maser's so-called ‘signal’ transition is sufficiently narrow, the maser action itself can provide the means of realizing an oscillator's frequency reference [(A) above], where the oscillator's output frequency is predominantly determined by the frequency of the said signal transition—acting as the reference—as opposed to the centre frequency of the maser's associated electromagnetic mode, whose linewidth is broader than that of the signal transition, with which the signal transition interacts. This possibility was in fact immediately appreciated upon the very first experimental demonstration of maser action [‘Molecular Microwave Oscillator and New Hyperfine Structure in the Microwave Spectrum of NH3’, J. P Gordon, H. J. Zeiger and C. Townes, Physical Review, 95, pp. 282-284 (1955)]. In such ‘atomic’ maser oscillators, frequency stability demands that the linewidth of the maser transition be as narrow as possible, i.e. that the transition's line Q be as high as possible. To reduce ‘pulling’ of the atomic maser oscillator's frequency, the Q of the electromagnetic mode with which the signal transition interacts should, on the other hand, be as low as is compliant with the conditions required for sustained, above-threshold active maser oscillation, at a reasonable output power. Here, the desired relative linewidths of the signal transition and the electromagnetic mode lie in stark contrast to Dick's SCMO, where the operating electromagnetic mode had a Q (˜109) that was approximately seven orders of magnitude higher than the line Q of the ruby's atomic (i.e. paramagnetic) signal transition (˜102). This fundamental ‘reversal of roles’ between the atomic maser transition and the electromagnetic resonance was indeed explicitly noted by Dick and Strayer, when comparing the workings of their SCMO to that of the (atomic) hydrogen maser. [See ‘The Superconducting Cavity Stabilized Ruby Maser Oscillator’, Dick, G. J. and D. M. Strayer, Proceedings of the Fifteenth Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, pp. 723-739, (1983).]
When describing maser oscillators, it is often necessary to distinguish between and quantify the various processes through which energy is lost from or gained by an electromagnetic mode. In this regard, is useful to define a mode's ‘non-magnetic’ Q as the Q that it would have were its interaction with the maser oscillator's paramagnetic ions (somehow) turned off.
As early as 1960, Bloembergen considered the realization of atomic maser oscillators of high frequency stability in solid-state systems (as opposed to those based on atomic/molecular beams). [See ‘The Zero-Field Solid State Maser as a Possible Time Standard’, in Quantum Electronics. A Symposium, Columbia University Press, New York, pp. 160-166 (1960).] He speculated, perhaps rather optimistically, that certain solid-state maser materials could exhibit paramagnetic transitions with linewidths almost as narrow as those exhibited by atoms in a vacuum. He also pointed out several advantages of those special solid-state systems (viz. the ‘zero-field masers’ previously mentioned) where maser action is possible without the application of a d.c. magnetic bias field.
Despite Bloembergen's imaginative early conjectures, by far the most successful atomic maser oscillator to date has been the (active) hydrogen maser oscillator, whose masering atoms compose a rarified (state-selected) gas, held within what is otherwise an evacuated bulb. Vanier provides a review [‘The Active Hydrogen Maser: State of the Art and Forecast’, J. Vanier, Metrologia, vol. 18, pp. 173-186 (1982)]. A conventional hydrogen maser operates at near-room temperature (the temperature of the hydrogen atoms emitted from the system's rf-dissociator is higher though the emitted atoms become quickly thermalized to the temperature of the walls of the maser's storage bulb.) The short-term stability (<100 s) of the hydrogen maser is limited by the low power of its output, which is typical no greater than −90 dBm or 1 pW. Its longer term frequency stability is limited by, amongst other processes, fluctuations in the dimensions of the maser's electromagnetic cavity.
Cryogenic or ‘cold’ hydrogen masers (CHMs), whose storage bulbs are coated with super-fluid helium, necessarily operate below the so-called lambda-point temperature for liquid helium at 2.17 K. They derive several benefits from their cryogenic operation: (i) lower thermal ‘Schawlow-Townes’ noise in the maser oscillator, (ii) lower thermal (Johnson) noise imparted by the (now potentially cryogenic) receiving amplifier, and (iii) a reduction in cavity pulling due to the lower thermal expansion, as well as lower change in the dielectric permittivity, of the materials that compose the maser's electromagnetic cavity. These significant benefits come, however, at the cost of other phenomena that significantly affect stability, most notably (the temperature and pressure dependence of) hyperfine spin-exchange, which cannot be compensated through (conventional) so-called ‘spin-exchange tuning’.
Fundamental Limits
The fundamental limits on a maser oscillator's short-term frequency stability are well understood by experts in the art of their design and construction. [See, for example, ‘Analysis of Fundamental and Systematic Effects Limiting the Hydrogen Maser Frequency Stability’, by E. Mattison and R. F. C Vessot, Proceedings of Twenty-first Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, Redondo Beach, Calif., USA, pp. 433-444 (1989)].
Thermal noise that is generated within and amplified by the maser oscillator causes fluctuations in the frequency of the maser oscillator's signal output. These frequency fluctuations can be described by their corresponding Allan deviation, as introduced above, which takes the form:
                                          σ            M                    ⁡                      (            τ            )                          =                              1                          Q              L                                ⁢                                                    k                ⁢                                                                  ⁢                                  T                  M                                                            2                ⁢                P                ⁢                                                                  ⁢                τ                                                                        (        1        )            where σM(τ) is the Allan deviation of the fractional frequency fluctuations of the maser oscillator's signal output as a function of the temporal sampling interval, τ; QL is the loaded, non-magnetic Q of the signal electromagnetic mode with which the maser oscillator's signal transition interacts, k is Boltzmann's constant, TM is the absolute temperature of the maser oscillator, and P is the overall rate at which energy is removed or dissipated from the maser signal mode due to both loading (i.e. the coupling out of signal power) and internal electromagnetic losses.
If the power of the maser oscillator's output signal is low, then the noise that is imparted onto it by its so-called ‘receiving amplifier’, which is generally necessary for boosting the power of the maser oscillator's output signal to a usable level, can adversely affect the frequency stability of the boosted signal. The Allan deviation that describes these additional frequency fluctuations takes the form:
                                          σ            M                    ⁡                      (            τ            )                          =                              1                          2              ⁢              π              ⁢                                                          ⁢                              f                M                            ⁢              τ                                ⁢                                                    3                ⁢                k                ⁢                                                                  ⁢                                  T                  R                                ⁢                B                                            2                ⁢                                  P                  0                                                                                        (        2        )            where σR (τ) is the Allan deviation of the additional fractional frequency fluctuations at the output of the receiving amplifier as a function of the sampling interval τ, and fM is the frequency of the maser oscillator's signal transition, TR is the so-called ‘noise temperature’ of the receiving amplifier, B is the effective noise bandwidth of the same, and P0 is the power of the signal delivered to the input of the receiving amplifier from the maser oscillator's output. P0 is some finite fraction of the total power dissipated in the maser's signal electromagnetic mode, P [as introduced in connection with equation (1) above]; the fraction P0/P is typically 0.25—though could be higher or lower depending on the application for which the maser oscillator's operating parameters are optimised. The receiving amplifier may also impart significant frequency flicker (phase) noise onto the boosted signal.
In general, the above two formulae indicate that it is advantageous with regard to frequency stability for the temperature, be it TM or TR (or both), to be low, and for the maser oscillator's operating power, be it P or P0 (or both), to be high. In these two regards, conventional hydrogen masers are limited by both their high operating temperature and, in particular, by their low output power.